WEAK-CONVERGENCE OF SEQUENCES OF FIRST PASSAGE PROCESSES AND APPLICATIONS

Suppose {X(n)}(n greater than or equal to 1) are stochastic processes all of whose paths are nonnegative and lie in the space of right conti nuous functions with finite left limits. Moreover, assume that X(n) (p roperly normalized) converges weakly to a process X, i.e., for some de terministic function mu and theta(n)-->0, theta(n)(-1)(X(n)-mu)-->(d)X . This paper considers the description of the weak limiting behavior o f the sequence of first passage processes <(X)over tilde (-1)(n)>(t)=i nf{x:<(X)over tilde (n)>(x)greater than or equal to t} where <(X)over tilde (n)>(x)=rho(nx)X(n)(x) and rho(.) is such that (X) over tilden$( x) has nondecreasing paths. We present a number of important motivatin g examples including empirical processes associated with U-statistics, empirical excursions above a given barrier, stopping rules in renewal theory and weak convergence in extreme value theory and point out the wide applicability of our result. Weak functional limit theorems for general quantile-type processes are derived. In addition, we investiga te the asymptotic behavior of integrated kernel quantiles and establis h: (i) an invariance principle; (ii) a strong law of large numbers; an d (iii) a Bahadur-type representation which has many consequences, amo ng which is a law of the iterated logarithm.